Contents lists available atScienceDirect
Topology
and
its
Applications
www.elsevier.com/locate/topol
Algebraic
entropy
on
strongly
compactly
covered
groups
Anna Giordano Brunoa,∗, Menachem Shlossbergb, Daniele Tollera
a
DipartimentodiScienzeMatematiche,InformaticheeFisiche,UniversitàdegliStudidiUdine,Viadelle Scienze206,33100Udine,Italy
bMathematicsUnit,ShamoonCollegeofEngineering,56BialikSt.,Beer-Sheva84100,Israel
a r t i cl e i n f o a b s t r a c t
Articlehistory:
Received24May2018
Receivedinrevisedform31October 2018
Accepted2November2018 Availableonline31May2019
Keywords:
Algebraicentropy Locallycompactgroup
Stronglycompactlycoveredgroup Compactlycoveredgroup FC-group
Locallyfinitegroup Continuousendomorphism Discretedynamicalsystem AdditionTheorem Limit-freeFormula BridgeTheorem
Weintroduceanewclassoflocallycompactgroups,namelythestronglycompactly covered groups, which are the Hausdorff topological groups G such that every elementof G is containedinacompactopennormalsubgroupof G. Forcontinuous endomorphisms φ : G → G of these groups we compute the algebraic entropy and study its properties. Also an Addition Theorem is available under suitable conditions.
©2019TheAuthors.PublishedbyElsevierB.V.Thisisanopenaccessarticle undertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
The topological entropyfor continuousselfmaps of compactspaces was introduced byAdler, Konheim and McAndrew [1], in analogy with the measure entropystudied in ErgodicTheory by Kolmogorovand Sinai.Lateron,Hood[23] extendedBowen-Dinaburg’sentropy(see[3,15])touniformlycontinuousselfmaps of uniform spaces. This notion of entropy coincides with the topological entropyfrom [1] in thecompact case,and it canbeconsideredinparticular for continuousendomorphisms φ of locally compact groupsG
(see[21]).
Alsothealgebraic entropyhasitsrootsinthepaper [1],where itwasconsideredforendomorphisms of (discrete) abelian groups. Weiss [32] studied this concept in the torsion case; in particular, he found the
* Correspondingauthor.
E-mailaddresses:anna.giordanobruno@uniud.it(A. Giordano Bruno),menacsh@sce.ac.il(M. Shlossberg), daniele.toller@uniud.it(D. Toller).
https://doi.org/10.1016/j.topol.2019.05.022
0166-8641/©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).
preciseconnectionofthealgebraicentropywiththetopologicalentropy(andalsowiththemeasureentropy) by means of Pontryagin duality,via a so-called Bridge Theorem.For a recent fundamentalpaper onthe algebraic entropyforendomorphismsof torsionabeliangroupswereferthereaderto[12].
Peters[26] definedthealgebraicentropydifferentlywithrespecttoWeissandhisdefinitionwasrestricted toautomorphismsofdiscreteabeliangroups.Peters’entropyneednotvanishontorsion-freeabeliangroups unlike thealgebraicentropyfrom[1].Atthesametime,these twonotionsofalgebraicentropycoincideon automorphismsoftorsionabeliangroups.In[11] Peters’definitionwasappropriatelymodifiedtointroduce algebraic entropy forendomorphisms ofdiscrete abelian groupsand allthefundamental propertiesofthe algebraic entropywereextended tothis generalsetting. TheBridge Theorem wasextendedto alldiscrete abeliangroupsin[7].Moreover,aconnectionwasfoundbetweenthealgebraicentropyandLehmer’sproblem fromNumberTheorybasedontheso-calledAlgebraicYuzvinskiFormula(see[19,20]),whichcomputesthe algebraic entropyofanendomorphismof afinite dimensionalrationalvectorspaceintermsoftheMahler measureofitscharacteristicpolynomial;itisthecounterpartoftheso-calledYuzvinskiFormula,provedby Yuzvinski [34],thatcomputesthetopologicalentropyofanendomorphismofafullsolenoidintermsofthe Mahler measure of itscharacteristic polynomial(clearly, theAlgebraic Yuzvinski Formula canbe derived from theYuzvinskiFormula,and viceversa,byusingtheBridgeTheorem).
Peters[27] extendedhisdefinitionofalgebraicentropyfrom[26] totopologicalautomorphismsoflocally compactabeliangroups,andVirili[30] modifiedthisdefinition(inasimilarmannerasin[11])tocontinuous endomorphismsoflocally compactabeliangroups.ABridgeTheoremisavailablefortopological automor-phisms of locally compact abelian groups (see [27,31]) and for continuous endomorphisms of compactly covered locally compact abelian groups (see [10]). Recall that a topological group G is called compactly covered if eachelement of G is containedinsomecompact subgroupof G.
ThecommutativityofthegroupsinVirili’sdefinitioncanbeomittedasitwasdescribedin[8].Wegive now thegeneraldefinition.
Inthispaper weconsideralwaysHausdorfftopologicalgroups.Foratopologicalgroup G, wedenoteby
C(G) the familyof all compactneighborhoods of theidentity element eG of G, and byEnd(G) the set of
continuous group endomorphisms of G. As usual we denote by N and N+ the natural numbers and the
positiveintegersrespectively.
Let G be alocallycompactgroupand μ be arightHaar measureon G. For φ ∈ End(G),asubset U of G, and n ∈ N+,the n-th φ-trajectory of U is
Tn(φ, U ) = U· φ(U) · . . . · φn−1(U ).
When U ∈ C(G), thesubset Tn(φ, U ) is also compact,so ithasfinite measure.The algebraic entropy of φ
with respect to U is
Halg(φ, U ) = lim sup n→∞
log μ(Tn(φ, U ))
n , (1.1)
and itdoesnotdependonthechoiceoftheHaarmeasure μ on G. The algebraic entropy of φ is halg(φ) = sup{Halg(φ, U ) : U∈ C(G)}.
Note that halg vanishesoncompactgroups.
Inthis paperwe studythealgebraic entropyofcontinuousendomorphisms φ of locallycompactgroups
G satisfying oneoftheequivalentconditionsstatedinProposition1.1 below;we callthese groups strongly compactly covered.
First we recall thatan FC-group is agroup inwhich everyelement hasfinitely many conjugates. It is known thatatorsion groupis anFC-groupif and onlyifeach of itsfinite subsets iscontained inafinite
normal subgroup (see [28, 14.5.8]);for this reason thetorsion FC-groups are alsocalled locally finite and normal. Inparticular, torsionFC-groupsarelocally finite.
Givenatopologicalgroup G, wedenotebyB(G) andN (G) thesubfamiliesofC(G) ofallcompactopen subgroupsandallcompactopennormalsubgroupsof G, respectively,andbyK(G) thefamilyofallcompact subsetsof G. Clearly,N (G)⊆ B(G)⊆ C(G)⊆ K(G).
Proposition1.1. Let G be a topological group. Then the following conditions are equivalent (and define G to be a strongly compactly covered group):
(1) G isa locally compact group and N (G) is cofinal in C(G);
(2) there exists K∈ N (G) such that G/K is a torsion FC-group;
(3) for every g∈ G there exists Ng∈ N (G) such that g∈ Ng;
(4) N (G) is cofinal in K(G).
Lookingatcondition(3)inProposition1.1,itisclearthatastronglycompactlycoveredgroupislocally compact and compactly covered. In view of condition (2), a strongly compactly covered group can also be called compact-by-(discrete torsion FC-group). In particular, compact groups are strongly compactly covered.
We proveProposition 1.1 in Section2, where we also discuss several propertiesof strongly compactly coveredgroups.Inparticular, theclassof stronglycompactlycovered groupsis stableundertaking closed subgroups, Hausdorffquotients andfinitedirectproducts (seeLemma2.3 andLemma2.4).Moreover, the stronglycompactlycovereddiscretegroupsarepreciselythetorsionFC-groups,whilethecompactlycovered discrete groupsareprecisely the torsiongroups(see Lemma2.2).As aconsequence, wesee thattheclass of strongly compactly covered groups is strictly contained in that of locally compact compactly covered groups,asforexamplethereexist torsiongroupsthatarenotFC-groups.
On the other hand, an abelian topological group G is strongly compactly covered precisely when it is locally compact and compactly covered (see Corollary 2.1);moreover, a locally compact abelian groupis compactly coveredprecisely when itsPontryagindual groupis totallydisconnected (see Remark6.3).So, someexamplesofstronglycompactly coveredabeliangroupsare:
(1) compactabeliangroups;
(2) locallycompacttorsionabeliangroups(inparticular, finiteabeliangroups); (3) the p-adic numbersQp.
Since the class ofstrongly compactly covered groupsis stable under taking finite directproducts, also QNp isstronglycompactlycovered.ThealgebraicentropyofcontinuousendomorphismsofQNp iscomputed
in detail in [19, Fact A] (its counterpart for the topological entropy can be found in [24]); this was a fundamentalstepintheproofoftheAlgebraicYuzvinski Formulain[19].
Nowrecallthefollowingproperty,thatmotivatedustostudythealgebraicentropyintheclassofstrongly compactlycoveredgroups,sinceitis clearfromthedefinition(seealso Remark3.3)thatwhencomputing the algebraic entropy it suffices to consider any cofinal subfamily of C(G). Clearly, if G is a topological abeliangroup,thenB(G)=N (G).
Fact 1.2.[10, Proposition 2.2] If G is a compactly covered locally compact abelian group, then B(G) is cofinalinC(G).
Generalizingthemeasure-freeformulagiven in[8] intheabeliancase,inLemma3.2we provethatif G
Halg(φ, U ) = lim n→∞
log[Tn(φ, U ) : U ]
n .
Incase G is stronglycompactlycovered,sinceN (G) iscofinalinC(G) byProposition1.1(1),weobtain
halg(φ) = sup{Halg(φ, U ) : U∈ N (G)}. (1.2)
These formulasallowfor asimplifiedcomputationof the algebraic entropyinstrongly compactlycovered groups.
In Section 4 we study the basic properties of the algebraic entropy for continuous endomorphisms of strongly compactlycoveredgroups:Invariance underconjugation,LogarithmicLaw, weakAddition Theo-rem, MonotonicityforclosedsubgroupsandHausdorffquotients.
In particular, the Invariance under conjugation (see Corollary 4.3) holds in the more general setting of continuous endomorphisms of locally compact groups, so the algebraic entropy is an invariant of the category offlows of locally compactgroups. Indeed,a flow of thecategory of locally compact groupsisa pair(G, φ), with G a locallycompactgroupand φ ∈ End(G);onecandefinethealgebraicentropyoftheflow (G, φ) as halg(φ).Amorphismbetweentwoflows (G, φ) and (G1, ψ) is acontinuousgrouphomomorphism α : G → G1 such that ψα = αφ; in case α is a topological isomorphism, the flows (G,φ) and (G1, ψ)
are isomorphic.Inviewof theInvarianceunderconjugation property, isomorphicflows(G, φ) and (G1, ψ)
havethesamealgebraic entropy halg(φ)= halg(ψ),thatis,flowswithdifferentalgebraicentropycannotbe
isomorphic.
We seethatthe algebraicentropy oftheidentity automorphism ofastrongly compactlycovered group is zero. This is not always true in the non-abelian case, in fact the identity automorphism of a finitely generated groupof exponentialgrowth hasinfinitealgebraic entropy(see[8,17]).
As afundamental example we compute the algebraic entropyof the shifts, showing inparticular that Invarianceunderinversionisnotavailableingeneralfortopologicalautomorphisms φ of stronglycompactly coveredgroups.Indeed,wefindthepreciserelationbetween halg(φ) and halg(φ−1) byusingthe modulus of
φ (see Proposition4.9).
The so-calledLimit-freeFormulasprovide awayto simplifythecomputationofboth thealgebraic and the topological entropyindifferent contexts. For example,Yuzvinski [34] provided such aformula forthe algebraic entropy of endomorphisms of discrete torsion abelian groups. However, it turned out that his formulaholdstrueforinjectiveendomorphisms[6,14] eventhoughitisnottrueingeneral(see[6,Example 2.1]). A general Limit-freeFormula forthe algebraic entropyinlocally finite groupsis given in[6], where also its counterpart for the topological entropy in totally disconnected compact groups is considered. A Limit-freeFormulaforthetopologicalentropyofcontinuousendomorphismsoftotallydisconnectedlocally compact groupsisgivenin[21],extendingaresultfrom [16] (seealso[5]).
Inspired by[4,21], wefindinProposition5.3aLimit-freeFormulaforthealgebraic entropythatcanbe described asfollows. Foralocally compactgroup G, φ ∈ End(G) and U ∈ N (G),weprovethat
Halg(φ, U ) = log[U− : φ−1U−],
where U− is the smallest (open) subgroup of G containing U and inversely φ-invariant (i.e., satisfying
φ−1U− ≤ U−). Note that U− is defined in [4] in the setting of locally linearly compact vector spaces, “dualizing” asimilarconstructionusedforthetopologicalentropyin[4,21],whichwasinspiredbyideasof Willis[33].
InSection6wecompareourformulastatedabovewiththeLimit-freeFormulaforthetopologicalentropy of continuous endomorphismsoftotally disconnected locally compactgroupsobtainedin[21]. This allows us toproduceanalternativeproof(seeTheorem 6.5)oftheBridgeTheoremgivenin[10].
Themain propertyofentropyfunctionsistheso-calledAdditionTheorem,thatmeansadditivityofthe entropyfunction(see Equation(1.3) below).It isalsoknownasYuzvinski’saddition formula,sinceitwas
first proved by Yuzvinski for the topological entropy in thecase of separable compact groups [34]. Later on,Bowen provedin[3, Theorem 19] aversionof theAdditionTheoremfor compactmetricspaces,while thegeneralstatementforcompactgroupsisdeducedfromthemetrizablecasebyDikranjanandSanchisin [13,Theorem8.3].TheAdditionTheoremplaysafundamentalrolealsointheUniquenessTheoremforthe topologicalentropyinthecategory ofcompactgroupsprovidedbyStoyanov[29].
TheAdditionTheorem forthetopologicalentropywasrecentlyextendedto continuousendomorphisms oftotallydisconnectedlocallycompactgroupsin[21] undersuitableassumptions,inparticularitholdsfor topologicalautomorphisms oftotallydisconnectedlocallycompactgroups.
AfirstAdditionTheoremforthealgebraicentropywasprovedin[12,Theorem3.1] intheclassofdiscrete torsion abelian groups, and waslater generalizedto the class of discrete abelian groupsin [11, Theorem 1.1].Notethatadiscreteabeliangroupisstronglycompactlycoveredpreciselywhenitistorsion.
Ontheotherhand,itisknown[17] thattheAdditionTheoremdoesnotholdingeneralforthealgebraic entropyevenfordiscrete solvablegroups(whileitsvalidityfornilpotentgroupsisanopen problem).This comes from the strict connection of the algebraic entropywith the groupgrowth from Geometric Group Theory(see[8,9,18]).
Let G be a locally compact group, φ ∈ End(G) and H a closed normal φ-invariant (i.e., satisfying
φ(H) ≤ H) subgroupof G. WesaythattheAdditionTheoremholdsforthealgebraic entropyif
halg(φ) = halg( ¯φ) + halg(φH), (1.3)
where φ H istherestrictionof φ to H and φ : G/H¯ → G/H istheinducedmaponthequotientgroup.
Weconsiderthefollowinggeneralproblem.NotethatitisnotevenknownwhethertheAdditionTheorem holdsingeneralforlocallycompactabeliangroups(see[8,10]).
Problem1.3. Forwhichlocally compactgroupsdoestheAdditionTheoremhold?
As our main result, we prove in Section7 that the AdditionTheorem holds for astrongly compactly coveredgroup G in case H is aclosednormal φ-stable (i.e., φ(H) = H)subgroupof G with ker φ≤ H.In particular, wefindthattheAdditionTheorem holds fortopologicalautomorphisms ofstrongly compactly coveredgroups(seeCorollary 7.2).
InthediscretecasethismeansthattheAdditionTheoremholdsforautomorphismsoftorsionFC-groups (thisisafirstextensionof[12,Theorem3.1] tosomenon-abeliangroups),andweconjecturethatthesame resultcanbe extendedtoalllocallyfinitegroups(seeConjecture7.3).
Another consequence of our Addition Theorem (see Corollary 7.7) is that, to compute the algebraic entropyof atopological automorphism φ of astrongly compactlycovered group G, onecanassume G to
be totallydisconnected;indeed,inthiscasetheconnectedcomponent c(G) of G is φ-stable and halg(φ)=
halg( ¯φ), whereφ : G/c(G) ¯ → G/c(G) isatopologicalautomorphism.
2. Stronglycompactlycoveredgroups
We start this section by proving Proposition 1.1, which characterizes the strongly compactly covered groups.
Proof of Proposition1.1. (1)⇒(2) Assume first that G is a locally compact group with N (G) cofinal in
C(G),and let H ∈ C(G).Bythecofinalityassumption,thereexists K∈ N (G) containing H.
Let X be a finite subset of the discrete group G/K containing the identity element, and consider the canonical projection q : G → G/K. Then, q−1(X) ∈ C(G), being afinite union of cosets of thecompact opensubgroup K, so thereexists M ∈ N (G) suchthat q−1(X)⊆ M.Itfollowsthat X⊆ q(M),and q(M )
is afinite normal subgroup of G/K, being compact inG/K discrete. This proves that G/K is a torsion FC-groupbyLemma2.2(3).
(2)⇒(3)Assume thatthere exists K ∈ N (G) such thatG/K is atorsion FC-group,let g ∈ G andlet
n ∈ N+ be the order of gK in G/K. Then g∩ K = gn,so K,g/K ∼=g/gn isfinite. As G/K is
a torsion FC-group there exists a finite normal subgroup S of G/K containing K, g/K. It follows that
g∈ Ng= q−1(S)∈ N (G),where q : G → G/K isthecanonicalprojection.
(3)⇒(4)Fix C∈ K(G) andletusshowthatthereexists N ∈ N (G) suchthat C⊆ N.Byourassumption, forevery c ∈ C thereexists Nc∈ N (G) with c ∈ Nc.Thecompactnessof C implies thatthereexist finitely
many c1, . . . , cn ∈ C suchthat C⊆
n
i=1Nci ⊆ Nc1· · · Nck.Sotake N = Nc1· · · Nck ∈ N (G).
(4)⇒(1)Firstofall, G is locallycompactasN (G) isnon-empty,andthethesisfollowsfromthe contain-mentC(G)⊆ K(G). 2
The followingis adirectconsequence ofProposition1.1for abeliangroups.Theimplication (5)⇒(1) is Fact1.2,while (3)⇒(5)istrivial.
Corollary 2.1. Let G be an abelian topological group. Then the following conditions are equivalent:
(1) G is a locally compact group and B(G) iscofinal in C(G);
(2) there exists K∈ B(G) such that G/K is torsion;
(3) for every g∈ G there exists Ng ∈ B(G) such that g∈ Ng;
(4) B(G) is cofinal in K(G);
(5) G is a locally compact compactly covered group.
In particular, G is strongly compactly covered if and only if G is locally compact and compactly covered.
By Corollary 2.1, a locally compact abelian group G is compactly covered precisely when the family
N (G)=B(G) iscofinalinC(G) (orinK(G)). Itis worthnotingthatincase G is non-abelian, B(G) need not be cofinal inC(G), even ifG is compactly covered. This follows for examplefrom Lemma2.2 below, taking into accountthatthereexist (necessarilynon-abelian)discrete torsiongroupswhicharenotlocally finite.
Foradiscretegroup G, notethat: (1) C(G)={X ⊆ G: X finite,eG ∈ X};
(2) B(G)={H ≤ G: H finite};
(3) N (G)={N ≤ G: N finite normal}. Lemma 2.2. Let G be a discrete group. Then:
(1) G is compactly covered if and only if it is torsion;
(2) B(G) is cofinal in C(G) if and only if G is locally finite;
(3) N (G) is cofinal in C(G) (i.e., G is strongly compactly covered) if and only if G is a torsion FC-group.
Proof. (1) Thisequivalencefollows fromthedefinitions.
(2) Let G be alocally finitegroupand X ∈ C(G).As G is locallyfinite,thefinitesubset X generates a finite subgroup F . Therefore, X ⊆ F ∈ B(G),andthisprovesthecofinalityofB(G) in C(G).
Conversely,assumethat G is notlocallyfinite.So,thereexistsafinitesubset F of G which generatesan infinitesubgroup. Withoutlossofgenerality,wemayassumethat eG∈ F ,sothat F ∈ C(G).Ontheother
(3)Thisimmediatelyfollowsfromtheequivalencebetweenthedefinitionoflocallyfiniteandnormaland thepropertyofbeing atorsionFC-group(see[28, 14.5.8]). 2
AsaconsequenceoftheabovelemmaandinviewofProposition1.1,weobtainthatthetorsionFC-groups arethestronglycompactlycoveredgroupswith respecttothediscretetopology.
ItiseasytoseethattheclassofdiscretetorsionFC-groupscontainsallfinitegroupsandisstableunder takingsubgroups,quotientsorformingdirectsums.Nowweextendthisresulttothelargerclassofstrongly compactlycoveredgroups.
Lemma 2.3. The class of strongly compactly covered groups is stable under taking closed subgroups and Hausdorff quotients.
Proof. Let G be a strongly compactly covered group, and letK ∈ N (G) be such that G/K is atorsion FC-group,byProposition1.1.
If H is aclosedsubgroupof G, then H∩ K ∈ N (H).Since H/H∩ K ∼= HK/K≤ G/K wededucethat
H/H∩ K is atorsionFC-group,so H is stronglycompactlycoveredbyProposition1.1.
Assumingthat H is alsonormal,onehas KH/H∈ N (G/H),beingtheimageof K in G/H. Astheclass ofdiscretetorsionFC-groupsisclosedundertakingquotients,wecompletetheproofusingtheisomorphisms
(G/H)/(KH/H) ∼= G/KH ∼= (G/K)/(KH/K) andagainapplying Proposition1.1. 2
Now we check that the class of strongly compactly covered groups is stable also under taking finite products.
Lemma2.4. Let G1, G2 be strongly compactly covered groups. Then
F = {U1× U2: Ui ∈ N (Gi)} ⊆ N (G1× G2) are cofinal subfamilies of C(G1× G2).
In particular, G1× G2 is a strongly compactly covered group.
Proof. Thegroup G1× G2 islocally compact,anditsufficestoprovethatF is cofinalinC(G1× G2).For i = 1,2 let πi : G1× G2 → Gi be the canonical projection. If F ∈ C(G1× G2), then πi(F ) ∈ C(Gi) for
i = 1,2,sothere is Ui∈ N (Gi) for i = 1,2 suchthat F⊆ π1(F )× π2(F )⊆ U1× U2. 2
3. Algebraicentropy
Inthesequel,foragroup G and φ ∈ End(G),if U is asubgroupof G, wesometimesusetheshortnotation
Tn for Tn(φ, U ), and the equalities Tn = U φ(Tn−1) and Tn = Tn−1φn−1(U ). In the following lemma we
study thelatterequalitywhen U commutes with φn(U ) forevery n ∈ N
+. Inparticular,if U is normalin G, weseethatevery Tn isasubgroupof G, normalin Tn+1.
Lemma 3.1. Let U be a subgroup of a group G and φ ∈ End(G). If φn(U )U = U φn(U ) for every n ∈ N
+, then:
(2) for every n ∈ N+,
φn(U )Tn= Tnφn(U ),
so Tn is a subgroup of G.
If U is normal in G (so the above condition is satisfied), then Tn is normal in T =
k∈NTk for every
n ∈ N+.
Proof. (1) Let k, m ∈ N andassumethat k≥ m.Letting n = k− m≥ 0 weget φn(U )U = U φn(U ) bythe
hypothesis, hence φk(U )φm(U )= φm(φn(U )U )= φm(U φn(U ))= φm(U )φk(U ).
(2) Clearly, T1 = U ≤ G,and φ(U )U = U φ(U ) = T2,so T2≤ G.We proceedby inductionon n ∈ N+.
Let n ≥ 2 andassumetheinductivehypothesis Tn−1φn−1(U )= φn−1(U )Tn−1.Then
Tnφn(U ) = Tn+1= U φ(Tn) = U φ(Tn−1φn−1(U )) = U φ(φn−1(U )Tn−1) =
= U φn(U )φ(Tn−1) = φn(U )U φ(Tn−1) = φn(U )Tn.
Therefore, Tn+1= Tnφn(U )≤ G.
Now weassumethat U is anormalsubgroup of G, and weprovethat Tn is normalin T , checkingthat
Tn is normalin Tn+m for every m ∈ N.To this aim wefix m ∈ N,we write Tn+m = Tnφn(Tm), and we
provebyinductionon n that φn(T
m) normalizes Tn.
For n = 1 thisfollowsfromthenormalityof T1= U in G. Let n ≥ 2,andfixanelement φn(u)∈ φn(Tm).
Write Tn = U φ(Tn−1), and assumethat φn−1(u)Tn−1 = Tn−1φn−1(u) holds bythe inductive hypothesis.
Then
φn(u)Tn = φn(u)U φ(Tn−1) = U φn(u)φ(Tn−1) = U φ(φn−1(u)Tn−1) =
= U φ(Tn−1φn−1(u)) = U φ(Tn−1)φn(u) = Tnφn(u). 2
Wenow provethatif U ∈ N (G), thenonecanavoidtheuseoftherightHaar measureinthedefinition of Halg(φ,U ) given in Equation (1.1), and that the limit superior becomes alimit. This generalizes the
measure-free formulagivenin[8] intheabeliancase.
Lemma 3.2. Let G be a locally compact group, φ ∈ End(G) and U ∈ N (G). Then
Halg(φ, U ) = lim n→∞
log[Tn(φ, U ) : U ]
n .
Proof. If U ∈ N (G), then Tn = Tn(φ, U ) ∈ B(G) by Lemma3.1(1), and in particular theindex [Tn : U ]
is finite. Using the fact that Tn is a disjoint union of cosets of U , and the properties of μ, we obtain
μ(Tn)= [Tn: U ]μ(U ),sothatlog μ(Tn)= log[Tn: U ]+ log μ(U ).Aslog μ(U ) doesnotdependon n, passing
to thelimitsuperior for n → ∞ weobtain
Halg(φ, U ) = lim sup n→∞
log[Tn(φ, U ) : U ]
n .
If tn = [Tn : U ] then tn dividestn+1,as U ≤ Tn ≤ Tn+1, and letβn = tn+1/tn = [Tn+1 : Tn]. Now we
show thatthesequenceofintegers{βn}n≥1 isweaklydecreasing.Indeed,
Inparticular,{βn}n≥1 stabilizes,so let n0∈ N,and β∈ N besuchthatforeveryn ≥ n0we have βn= β,
i.e. tn= βn−n0tn0.
Then,
Halg(φ, U ) = lim sup n→∞ log βn−n0t n0 n = log β = limn→∞ log tn n . 2 (3.1)
Remark3.3.If G is alocallycompactgroup,and φ ∈ End(G),thentheassignment Halg(φ,−) ismonotone.
Indeed,if U ⊆ V areelementsofC(G), then Tn(φ,U ) ⊆ Tn(φ,V ), so Halg(φ, U ) ≤ Halg(φ, V ). So,
halg(φ) = sup{Halg(φ, U ) : U ∈ B}
foreverycofinalsubfamily B ofC(G).
If G is astrongly compactly covered group,then by Remark3.3 we obtainEquation (1.2). The latter generalizes[10, Theorem2.3(b)],whichdealswiththeabeliancase.
NotethatanequivalentformulationofLemma2.3isthatN (H) iscofinalinC(H) (respectively,N (G/H) is cofinalinC(G/H)) when H is a closed (respectively,closed and normal)subgroup of G. Thefollowing lemma presents smaller cofinalsubfamilies of C(H) and C(G/H) inthe samesetting. It has akey role in provingTheorem 7.1inviewofRemark3.3.
Lemma3.4. Let G be a strongly compactly covered group, and let H be a closed subgroup of G. Then:
(1) the family NG(H)={U ∩ H : U ∈ N (G)} is cofinal in C(H);
(2) if H is also normal in G, then the family NG(G/H)={πU : U ∈ N (G)} is cofinal in C(G/H), where
π : G → G/H is the canonical projection.
Proof. By Lemma 2.3, it suffices to prove that NG(H) is cofinal in N (H), and NG(G/H) is cofinal in
N (G/H).
Considerthecompactopen normalsubgroups K and Ng of G from Proposition1.1.
(1) LetV ∈ N (H).Then V K∈ B(G), andby ourassumption onG there exists U ∈ N (G) containing V K. Thus, U∩ H ≥ V K ∩ H ≥ V ∩ H = V .
(2)Let V ∈ N (G/H). Forevery v∈ V ,let u ∈ π−1v, so thatK, u∈ B(G) andK, u⊆ Nu ∈ N (G),
and v∈ π(Nu)∈ N (G/H).Bythecompactnessof V , thereexist u1, . . . , un∈ G suchthat V ⊆ni=1π(Nui).
Then U = Nu1Nu2· · · Nun∈ N (G), and V ≤ πU. 2
InthenotationofLemma3.4,applying Remark3.3we obtainthefollowingresult.
Corollary 3.5. Let G be a strongly compactly covered group, φ ∈ End(G), and H be a closed φ-invariant subgroup of G. Then:
(1) φH∈ End(H), and
halg(φH) = sup{Halg(φ, U ) : U ∈ NG(H)}.
(2) if H is also normal, and φ : G/H¯ → G/H denotes the induced map, then φ¯∈ End(G/H), and halg( ¯φ) = sup{Halg(φ, U ) : U ∈ NG(G/H)}.
4. Basicproperties
Inthissectionwelistthebasicpropertiesof halgofendomorphismsofstronglycompactlycoveredgroups.
Westartbyshowingthattheidentitymapof suchgroupshaszeroalgebraic entropy.
Lemma 4.1. If G is a locally compact group, then Halg(idG, U ) = 0 for every subgroup U ∈ C(G). In
particular, if G is strongly compactly covered, then halg(idG)= 0.
Proof. Obviously, ifU ∈ C(G) isasubgroup,then Tn(idG, U ) = U forevery n ∈ N+, so itsmeasure does
notdependon n, and Halg(idG, U ) = 0.
If G is strongly compactly covered, taking thesupremum over U ∈ N (G), we obtainhalg(idG)= 0 by
Equation(1.2). 2
Invarianceunderconjugationwasprovedingeneralforendomorphismsoflocallycompactabeliangroups byVirili[30, Proposition2.7(1)].This propertyholds truewithouttheabelian assumptionas notedin[8]. For thereader’s convenience weprovide aproof of this property inCorollary4.3 using thefollowing easy lemma.
Lemma 4.2. Let α : G → G1 be a group homomorphism, and φ ∈ End(G), ψ ∈ End(G1) be such that ψα = αφ. Then Tn(ψ, α(K)) = α(Tn(φ, K)) holds for every subset K of G, and for every n ∈ N.
Proof. If K is asubsetof G and n ∈ N, then
Tn(ψ, α(K)) = α(K)· ψα(K) · · · ψn−1α(K) = α(K)· αφ(K) · · · αφn−1(K) =
= α(K· φ(K) · · · φn−1(K)) = α(Tn(φ, K)). 2
Corollary 4.3 (Invariance under conjugation). Let α : G → G1 be a topological isomorphism of locally compact groups. If φ ∈ End(G) andψ = αφα−1, then Halg(φ,K) = Halg(ψ, α(K)) for every K∈ C(G). In
particular, halg(φ)= halg(ψ).
Proof. Let μ be aright Haarmeasure onG, anddefine aright Haarmeasure μ on G1 byletting μ(E)= μ(α−1(E)) foreveryBorelsubset E≤ G1.If K∈ C(G),then α(K) ∈ C(G1) and Tn(ψ, α(K)) = α(Tn(φ,K))
by Lemma 4.2. It follows that μ(Tn(φ, K)) = μ(α(Tn(φ, K))) = μ(Tn(ψ,α(K))), so Halg(φ, K) =
Halg(ψ,α(K)). Since α−1 isalsoanisomorphism,wededucethat halg(φ)= halg(ψ) bydefinition. 2
NowweprovetheLogarithmicLawforthealgebraicentropy,withrespecttoendomorphismsofstrongly compactly coveredgroups.
Lemma 4.4 (Logarithmic Law). Let G be a strongly compactly covered group and φ ∈ End(G). Then halg(φm)= m· halg(φ) for every m ∈ N.
Proof. Since theassertionisclearwhen m = 0 (seeLemma4.1),we mayfix m > 0. Letusshowfirstthat
halg(φm)≤ m· halg(φ). If n ∈ N+ and U ∈ N (G),then U ≤ Tn(φm, U ) ≤ Tmn−m+1(φ,U ). This implies
that Halg(φm, U ) = lim n→∞ log[Tn(φm, U ) : U ] n ≤ ≤ lim n→∞ log[Tmn−m+1(φ, U ) : U ] mn− m + 1 · limn→∞ mn− m + 1 n = mHalg(φ, U ),
Toprovetheconverseinequality,let U ∈ N (G). Observethat
W = Tn(φm, Tm(φ, U )) = Tnm(φ, U ),
soboth Tm(φ,U ) ∈ B(G) and W ∈ B(G) byLemma3.1(1), and U ≤ Tm(φ,U ) ≤ W .Then
halg(φm)≥ Halg(φm, Tm(φ, U )) = lim sup n→∞ log μ(Tn(φm, Tm(φ, U ))) n = = lim sup n→∞ log μ(Tnm(φ, U )) n = mHalg(φ, U ).
As the above inequality holds for every U ∈ N (G), this proves that halg(φm) ≥ m· halg(φ), and thus
halg(φm)= m· halg(φ). 2
Given twogroupendomorphisms φi : Gi → Gi for i = 1,2,inthefollowingproposition weconsider the
groupendomorphism φ1× φ2: G1× G2→ G1× G2,mapping(x, y) → (φ1(x),φ2(y)).
Proposition 4.5 (Weak Addition Theorem). Let G1, G2 be strongly compactly covered groups, and let φ1 ∈
End(G1), φ2∈ End(G2). Then halg(φ1× φ2)= halg(φ1)+ halg(φ2).
Proof. ByLemma2.4,thefamily{U1×U2: Ui∈ N (Gi)} iscofinalinC(G1×G2),and G1×G2isastrongly
compactlycoveredgroup.
For i = 1,2 let Ui∈ N (Gi),and n ≥ 1 beaninteger.Then
W = Tn(φ1× φ2, U1× U2) = Tn(φ1, U1)× Tn(φ2, U2),
so
[W : U1× U2] = [Tn(φ1, U1) : U1]· [Tn(φ2, U2) : U2].
Applyinglog,dividingby n, and takingthelimitfor n → ∞ weconcludethat
Halg(φ1× φ2, U1× U2) = Halg(φ1, U1) + Halg(φ2, U2). (4.1)
Now we take the suprema over U1 ∈ N (G1) and U2 ∈ N (G2) in Equation (4.1). Then the left hand
side gives halg(φ1× φ2) by Lemma2.4. Taking into account that Halg(φ1, U1) only depends on U1, while Halg(φ1, U2) onlydependson U2,therighthandsideof Equation(4.1) gives
sup{Halg(φ1, U1) + Halg(φ1, U2) : Ui∈ N (Gi), i = 1, 2} =
= sup{Halg(φ1, U1) : U1∈ N (G1)} + sup{Halg(φ2, U2) : U2∈ N (G2)} = halg(φ1) + halg(φ2). 2
We briefly mention here the monotonicity property of halg for endomorphisms of strongly compactly
coveredgroups,thatwillimmediately followfrom thestrongerresultprovedinProposition7.5.
Proposition 4.6 (Monotonicity). Let G be a strongly compactly covered group, φ ∈ End(G), H a closed normal φ-invariant subgroup of G, and φ : G/H¯ → G/H the induced map. Then
Let F be afinitediscretegroupandlet G = Gc⊕ Gd,with Gc= 0 n=−∞ F and Gd= ∞ n=1 F.
Endow Gd with the discrete topologyand Gc with the (compact)product topology,so both of them are
stronglycompactlycovered(for Gd,applyLemma2.2(3)).Endowedwiththeproducttopology, G is locally
compact, anditisalsostronglycompactlycoveredbyLemma2.4. The left shift is
Fβ : G→ G, (xn)n∈Z → (xn+1)n∈Z,
while the right shift is
βF: G→ G, (xn)n∈Z → (xn−1)n∈Z.
Clearly, βF andFβ are topological automorphismsof G such thatFβ−1 = βF.
Inthenextexamplewecomputethealgebraic entropyofthese shifts.
Example 4.7. Considerthegroup G =0n=−∞F⊕∞n=1F introduced above.Weverifynowthat
halg(Fβ) = 0 and halg(βF) = log|F |.
For k∈ N+,let Uk= 0 n=−∞ F × k n=1 F ∈ N (G).
LetuscheckthatthefamilyU = {Uk : k∈ N+} iscofinalinK(G).Indeed,let H ∈ K(G),andconsiderthe
canonical projection πd : G→ Gd.Then πd(H)⊆ Gd isfinite,so πd(H)⊆kn=1F for somek∈ N+, and H ⊆ Uk.
Clearly,
. . .≤Fβn(Uk)≤ . . . ≤Fβ(Uk)≤ Uk ≤ βF(Uk)≤ . . . ≤ βFn(Uk)≤ . . . .
Forall n ∈ N+, Tn(Fβ, Uk)= Uk, so
Halg(Fβ, Uk) = 0.
Ontheother hand,forevery n ∈ N+,
logTn+1(βF, Uk) Tn(βF, Uk) = log β n F(Uk) βFn−1(Uk) = logβF(Uk) Uk = log|F |, hence Halg(βF, Uk) = log|F |
The above example shows in particular that a topological automorphism φ of a strongly compactly covered group and its inverseφ−1 can havedifferent algebraic entropy. Nextwe givethe precise relation between halg(φ) and halg(φ−1),whichextends[30,Proposition2.7(3)] tosomenon-abeliangroups.
For a locally compact group G, let Aut(G) denote the group of topological automorphisms of G. If
μ is a left Haar measure on G, the modulus is a group homomorphism ΔG : Aut(G) → R+ such that μ(φE) = ΔG(φ)μ(E) forevery φ ∈ Aut(G) and everymeasurable subset E of G (see [22]). If G is either
compactordiscrete, thenΔG(φ)= 1 for every φ ∈ Aut(G).WealsodenoteΔG simplybyΔ.
Lemma4.8. Let G be a locally compact group, φ ∈ Aut(G), and U ∈ N (G). Then
(1) Δ(φ)= μ(φ(U ))μ(U ) = [U φ(U ):φ(U )][U φ(U ):U ] .
(2) If V is a compact subgroup of G and V ⊇ Uφ(U),then [V : U ]= [V : φ(U )]· Δ(φ).
Proof. (1) Since U and φ(U ) are compact, and U ∩ φ(U) is open, it follows that[φ(U ) : U ∩ φ(U)] and [U : U∩ φ(U)] arefinite.Therefore, μ(φ(U )) = [φ(U ): U∩ φ(U)]· μ(U ∩ φ(U)) and μ(U ) = [U : U∩ φ(U)]·
μ(U∩ φ(U)).Hence,
Δ(φ) = μ(φ(U ))
μ(U ) =
[φ(U ) : U ∩ φ(U)] [U : U∩ φ(U)] .
Moreover, since U is normal inG we have that U φ(U ) is asubgroup of G, and so [φ(U ) : U ∩ φ(U)] = [U φ(U ): U ] and[U : U∩ φ(U)]= [U φ(U ): φ(U )].
(2)Wehavethat
[V : U ] = [V : U φ(U )][U φ(U ) : U ] and
[V : φ(U )] = [V : U φ(U )][U φ(U ) : φ(U )]. Therefore,
[V : U ] = [V : φ(U )] [U φ(U ) : U ] [U φ(U ) : φ(U )], andtheconclusionfollowsfromitem (1). 2
Proposition4.9. Let G be a locally compact group, φ ∈ Aut(G) and U ∈ N (G).Then Halg(φ−1, U ) = Halg(φ, U )− log Δ(φ).
In particular, if G is a strongly compactly covered group, then halg(φ−1) = halg(φ)− log Δ(φ).
Proof. Let U ∈ N (G). As in the proof of Lemma 3.2, for every n ∈ N+ let tn = [Tn(φ, U ) : U ] and
t∗n = [Tn(φ−1, U ) : U ]. Moreover, Halg(φ,U ) = log β and Halg(φ−1, U ) = log β∗, where β and β∗ are
respectivelythevaluesat whichthesequences βn= tn+1tn and βn∗= t∗n+1
t∗n stabilize(seeEquation(3.1)).
Let n ∈ N+.ByLemma3.1 wehavethat
is acompactsubgroupof G. Since φn−1 isanautomorphism,Lemma4.8(2)gives t∗n= [Tn(φ−1, U ) : U ] = [φn−1(Tn(φ−1, U )) : φn−1(U )] = = [Tn(φ, U ) : φn−1(U )] = [Tn(φ, U ) : U ]· 1 Δ(φn−1) = tn· 1 Δ(φn−1).
Therefore, sinceΔ isahomomorphism, foreverysufficientlylarge n ∈ N+,
β = tn+1 tn = t ∗ n+1 t∗n · Δ(φn) Δ(φn−1) = β ∗· Δ(φ).
Then, log β = log β∗+ log Δ(φ),thatis,thefirstassertionoftheproposition.
Thesecond equalityinthethesisfollowsfrom thefirstonetaking thesupremumfor U ∈ N (G) inview of Equation(1.2). 2
From Example 4.7, where we computed that halg(Fβ) = 0 and halg(βF) = log|F |, and from
Proposi-tion4.9,itfollowsthatΔ(Fβ) =|F |−1 andΔ(βF)=|F |.
5. Limit-freeformula
Inthissectionweprovetheso-calledLimit-freeFormulaforthealgebraicentropy.Westartintroducing thefollowing usefulsubgroups.
Definition 5.1.Foralocally compactgroup G, φ ∈ End(G) and U ∈ N (G), let: (1) U(0)= U ;
(2) U(n+1)= U φ−1(U(n)) forevery n ∈ N;
(3) U−=n∈NU(n).
It iseasy to proveby inductionthatU(n+1) = φ−1(U(n))U isanopen normalsubgroup of G such that U(n)≤ U(n+1)forevery n ∈ N,sothatalso U−isanopennormalsubgroupof G for every U ∈ N (G).We
collectsomeofthepropertiesof U−.
Lemma 5.2. Let G be a locally compact group, φ ∈ End(G) and U ∈ N (G). Then:
(1) φ−1(U−)≤ U−;
(2) if H ≤ G issuch that U ≤ H and φ−1(H)≤ H, then U−≤ H;
(3) U−= U φ−1(U−);
(4) the index [U−: φ−1(U−)]= [U : U∩ φ−1(U−)] is finite;
(5) for every n ∈ N+, φ−n(T
n)= φ−1(U(n−1)).
Proof. (1) Followsfrom thefactthat φ−1(U(n))≤ U(n+1)forevery n ∈ N.
(2) It suffices to show thatU(n) ≤ H for every n ∈ N. If n = 0, then U = U(0) ≤ H, while for the inductive step,use φ−1(U(n))≤ φ−1(H)≤ H.
(3) Wehavealreadynotedabovethat U(n)≤ U(n+1) forevery n ∈ N,so U φ−1(U−) = U φ−1( n∈N U(n)) = U n∈N φ−1(U(n)) = n∈N U φ−1(U(n)) = n∈N U(n+1)= U−.
(4)Usingproperty (3)weobtain
U−/φ−1(U−) = (U φ−1(U−))/φ−1(U−) ∼= U/(U ∩ φ−1(U−)),
andthus[U−: φ−1(U−)]= [U : U∩ φ−1(U−)].Thequotient U/(U∩ φ−1(U−)) isfinite,since U∩ φ−1(U−) isanopensubgroupofthecompactgroup U , as φ−1(U−) isopenin G.
(5)For n = 1 theassertionfollows fromtheequalities T1= U = U(0).Weprovetheassertionfor n + 1
assumingthatitholdstruefor n ∈ N+.
Tosee firstthatφ−n−1(Tn+1)⊆ φ−1(U(n)),fix x ∈ φ−n−1(Tn+1).So, φn+1(x)∈ Tn+1= Tnφn(U ).This
impliesthatthere exists y ∈ U suchthat φn+1(x)∈ T
nφn(y),thatis, φn(φ(x)y−1)∈ Tn. Bytheinductive
hypothesis, φ(x)y−1∈ φ−n(Tn)= φ−1(U(n−1)).Hence,
φ(x) = (φ(x)y−1)y∈ φ−1(U(n−1))U = U φ−1(U(n−1)) = U(n).
Thismeansthat x ∈ φ−1(U(n)),asneeded.
Fortheconverseinclusionlet x ∈ φ−1(U(n)).Bytheinductivehypothesis,
φ(x)∈ U(n)= U φ−1(U(n−1)) = U φ−n(Tn).
Itfollows that
φn+1(x) = φn(φ(x))∈ φn(U )Tn= Tn+1,
andthus, x ∈ φ−n−1(Tn+1). 2
Inthefollowingresult,we generalizetheLimit-freeFormulaforthealgebraic entropyofendomorphism oftorsionabeliangroups,from[6,34].
Proposition5.3 (Limit-free Formula). Let G be a locally compact group. If φ ∈ End(G) and U ∈ N (G), then Halg(φ, U ) = log[U−: φ−1(U−)].
Proof. ByEquation(3.1), Halg(φ, U ) = log β,where β = βn= [Tn+1: Tn] for n ≥ n0 forsome n0∈ N.So,
itsufficestoshow that β = [U− : φ−1(U−)]. Forevery n ∈ N,
U∩ φ−1(U ) = U∩ φ−1(U(0))≤ U ∩ φ−1(U(n))≤ U ∩ φ−1(U(n+1))≤ U, sothesequenceof theindices {[U : U ∩ φ−1(U(n))]}
n∈N isweaklydecreasing,henceitstabilizes. Itfollows
that the sequence {U ∩ φ−1(U(n))}n∈N of subgroups of U also stabilizes, and let n1 ∈ N be such that U∩ φ−1(U(n))= U∩ φ−1(U(n1)) forevery n ≥ n 1. For m ≥ n1 wehave U∩ φ−1(U(m)) = n∈N (U∩ φ−1(U(n))) = U∩ n∈N φ−1(U(n)) = U∩ φ−1( n∈N U(n)) = U∩ φ−1(U−).
Therefore,forevery n ≥ max{n0, n1},Lemma5.2(4)gives
[U−: φ−1(U−)] =[U : U∩ φ−1(U−)] = [U : U∩ φ−1(U(n))] =
To conclude,weshow that[U(n): φ−1(U(n−1))]= [T
n+1: Tn]= β.Observethat
U(n)/φ−1(U(n−1)) = (U φ−1(U(n−1)))/φ−1(U(n−1)),
and that Tn+1/Tn= (φn(U )Tn)/Tn. Considerthemapping
Φ : (U φ−1(U(n−1)))/φ−1(U(n−1))→ (φn(U )Tn)/Tn, xφ−1(U(n−1)) → φn(x)Tn.
ByLemma5.2(5),Φ iswell-definedandinjective,andclearlyΦ isasurjectivehomomorphism. 2 ThefollowingresultwasinspiredbyFact6.4(provedin[21])whichisasimilarresultforthetopological entropy.
Proposition 5.4. Let G be a strongly compactly covered group, and φ ∈ End(G). Then
halg(φ) = sup{log[A : φ−1(A)] : A G, A open, φ−1(A)≤ A, [A : φ−1(A)] <∞} =: s.
Proof. Using Proposition5.3weobtain
halg(φ) = sup{Halg(φ, U ) : U ∈ N (G)} = sup{log[U−: φ−1(U−)] : U ∈ N (G)}.
Then halg(φ)≤ s,as forevery U ∈ N (G), U− is anopennormalsubgroupof G such that φ−1(U−)≤ U−
and [U−: φ−1(U−)] <∞ byLemma5.2(4).
Fortheconverseinequality,fixanopennormalsubgroup A of G such that φ−1(A)≤ A and[A: φ−1(A)] < ∞. Ouraimisto find U ∈ N (G) such that[U−: φ−1(U−)]≥ [A: φ−1(A)]. Since[A: φ−1(A)] <∞, there existsafinitelygeneratedsubgroup F ≤ A suchthat A = φ−1(A)F .Useitem(4)ofProposition1.1,tofind
K∈ N (G) suchthat F ≤ K.Weclaimthat
A = φ−1(A)U, (5.1)
where U = A ∩ K ∈ N (G).Firstobservethat
F ⊆ A ∩ K = U ⊆ A.
Takingalso intoaccountthat φ−1(A)≤ A weobtain
A = φ−1(A)F ⊆ φ−1(A)U ⊆ φ−1(A)A = A, whichproves Equation(5.1).
We nowshow that[U− : φ−1(U−)]≥ [A: φ−1(A)]. Since U ≤ A and φ−1(A)≤ A,Lemma5.2(2)gives
U− ≤ A,so φ−1(U−)≤ φ−1(A)≤ A.Now byEquation(5.1),
[U φ−1(U−) : φ−1(U−)]≥ [Uφ−1(U−)φ−1(A) : φ−1(U−)φ−1(A)] = [A : φ−1(A)]. Finally,[U−: φ−1(U−)]= [U φ−1(U−): φ−1(U−)] byLemma5.2(3). 2
6. Bridgetheorem
Foralocally compact abeliangroup G, itsPontryagin dualG is thegroupof allcontinuous homomor-phism from G to thetorus T ,equipped withthepointwise operation,and thecompactopen topology;by Pontryaginduality,G is atopologicalgroupsuchthat G iscanonically topologicallyisomorphic to G (see
[25]).
Forasubgroup H of G, theannihilatorof H in G is H ⊥={χ∈ G : χ(H) = 0},whileforasubgroup A
ofG, theannihilatorof A in G is A ={x∈ G: χ(x)= 0 for every χ∈ A}.
Then, for everyclosed subgroup H of G, we have(H⊥) = H,while H ≤ G is compact ifand only if
H⊥≤ G is open.Inparticular, H ∈ B(G) ifandonlyif H⊥ ∈ B( G).
Finally,if{Hi}i∈I isafamilyofopensubgroupsof G, then
i∈I Hi ⊥ = i∈I Hi⊥ and i∈I Hi ⊥ = i∈I Hi⊥.
Inwhatfollows, weneedalsothefollowingstandardpropertiesofPontryaginduality.
Lemma6.1. Let B≤ A be closed subgroups of a locally compact abelian group G. Then A/B is topologically isomorphic to B⊥/A⊥.
For a continuous group homomorphism φ : G → H, its dual is the continuous group homomorphism ˆ
φ : H → G, mapping f → f ◦ φ.
Lemma 6.2. Let G be a locally compact abelian group, φ ∈ End(G) and U be a closed subgroup of G. Then φ−1(U )⊥= φ(U⊥).
Remark6.3.Foratopological abeliangroup G, thesubgroup
B(G) ={K ≤ G : K compact}
isthebiggestcompactlycoveredsubgroupof G, so G is compactlycoveredexactlywhen B(G) = G.When
G is locally compact abelian, as B(G)⊥ = c( G), onehas that G is compactly covered ifand only if G is
totallydisconnected.
ThefollowingresultisaconsequenceoftheLimit-freeFormulaforthetopologicalentropyprovedin[21, Proposition3.9].
Fact 6.4. [21, Equation (3.10)] Let G be atotally disconnected locally compact group and φ ∈ End(G).
Then
htop(φ) = sup{log[φ(M) : M] : M ≤ G, M compact, M ≤ φ(M), [φ(M) : M] < ∞}.
Using Proposition5.4 andFact6.4,which arerespectively consequencesofthe Limit-freeFormulas for the algebraic and forthe topological entropy,we give inTheorem 6.5a newand very short proof for the BridgeTheorem from[10].
Theorem6.5. Let G be a compactly covered locally compact abelian group and φ ∈ End(G). Then halg(φ)=
Proof. ObservethatG is totallydisconnectedbyRemark6.3.
Let A be asubgroup of G. Thenthefollowingconditionsareequivalent: (i) A isopen, φ−1(A)⊆ A and A/φ−1(A) isfinite;
(ii) M = A⊥ iscompact, M⊆ φ(M ) and φ(M )/M is finite.
Undertheseconditions,byLemma6.2andLemma6.1respectively,
φ(M )/M = φ−1(A)⊥/A⊥ ∼=A/φ−1(A),
whichisisomorphicto A/φ−1(A),beingfinite.NowthethesisfollowsfromFact6.4andProposition5.4. 2 7. AdditionTheorem
This sectionisdedicatedto thefollowingAdditionTheorem.
Theorem7.1. Let G be a strongly compactly covered group, φ ∈ End(G), H a closed normal φ-stable subgroup of G with ker φ≤ H, and φ : G/H¯ → G/H the induced map. Then
halg(φ) = halg( ¯φ) + halg(φH).
As animmediate corollaryweobtainthefollowing resultforatopologicalautomorphism φ ∈ Aut(G) of G.
Corollary7.2. If G is a strongly compactly covered group, φ ∈ Aut(G) and H is a normal φ-stable subgroup of G, then
halg(φ) = halg( ¯φ) + halg(φH).
Inparticular,theAdditionTheoremholdsforautomorphismsofdiscretetorsionFC-groups.Thesegroups are locallyfinite,andweconjecture theAdditionTheorem holdsalsointhebiggerclassofdiscrete locally finite groups.
Conjecture7.3. If G is a discrete locally finite group, φ ∈ Aut(G) and H is a normal φ-stable subgroup of G, then
halg(φ) = halg( ¯φ) + halg(φH).
Inthesequel, G is astronglycompactlycoveredgroup, φ ∈ End(G),and H is aclosednormal φ-invariant
subgroup of G. Wedenote byπ : G → G/H the canonicalprojection, and by φ : G/H¯ → G/H the map inducedby φ on thequotientgroup.Obviously,φπ = πφ.¯
Firstweprovethefollowing easycharacterization ofthesubgroups H of G that we consider.
Lemma 7.4. Let G be a group, φ ∈ End(G) and H a closed subgroup of G. The following assertions are equivalent:
(1) H is φ-stable and contains ker(φ); (2) φ−1(H)= H andH ≤ Im(φ).
Proof. (1)⇒(2) Since H is φ-stable, we haveH = φ(H) ≤ φ(G)= Im(φ) and φ−1(H) = φ−1(φ(H)). As ker(φ)≤ H, φ−1(H)= φ−1(φ(H))= H.
(2)⇒(1) Since ker(φ) ≤ φ−1(H), the condition φ−1(H) = H implies that ker(φ) ≤ H and also that
φ(φ−1(H))= φ(H).As H ≤ Im(φ), φ(φ−1(H)) coincidesalsowith H. Itfollowsthat H = φ(H). 2
Nowwe provethefirst halfof theAdditionTheorem.Note thatitholdsinamoregeneralsetting than Theorem7.1.
Proposition 7.5. Let G be a strongly compactly covered group, φ ∈ End(G), H a closed normal φ-invariant subgroup of G, and φ : G/H¯ → G/H the induced map. Then
halg(φ)≥ halg( ¯φ) + halg(φH).
Proof. For U ∈ N (G), and n ∈ N,let Tn= Tn(φ,U ). Then U ≤ Tn∩ (UH)≤ Tn,so that
[Tn : U ] = [Tn: Tn∩ UH][Tn∩ UH : U], (7.1)
andwestudy separatelythetwoindicesintherighthandsideoftheaboveequation.
Let a = [Tn : Tn∩ UH] andconsider thefollowingHasse diagram inthelatticesof subgroupsof G and
of G/H. π G G/H TnH = Tn(φ, U H) Tn( ¯φ, π(U )) a a U H π(U ) Tn a Tn∩ (UH) U H {0}
Notethat TnH = Tn(φ, U H), since φH ≤ H andboth U and H are normalin G, so
a = [TnH : U H] = [Tn(φ, U H) : U H].
Moreover, both Tn(φ, U H) and U H contain H = ker π, so considering their images in G/H we have
[Tn(φ, U H) : U H] = [π(Tn(φ, U H)) : π(U H)], and the latter coincides with [Tn( ¯φ, π(U )) : π(U )] by
Lemma4.2.
To study the second index in the right hand side of Equation (7.1), let b = [Tn ∩ (UH) : U ]. As
Tn∩ (UH)= (Tn∩ H)U bythemodularlaw,wehave
where thelastinequalityfollowsfromtheinclusions Tn∩ H ≥ Tn(φ,U ∩ H)≥ U ∩ H. Tn∩ (UH) = (Tn∩ H)U b U Tn∩ H H Tn(φ, U∩ H) U∩ H b
Finally,from Equation(7.1) andtheabovediscussion,itfollows that
[Tn : U ]≥ [Tn( ¯φ, π(U )) : π(U )][Tn(φ, U∩ H) : U ∩ H].
Applying log,dividingby n, andtakingthelimitfor n → ∞ weconcludethat,forevery U ∈ N (G),
Halg(φ, U )≥ Halg( ¯φ, π(U )) + Halg(φH, U∩ H). (7.2)
Let U1, U2 ∈ N (G), and U = U1U2 ∈ N (G). Then π(U ) ≥ π(U1) are elements of N (G/H), and U ∩ H ≥ U2∩ H areinN (H),sothat
Halg( ¯φ, π(U ))≥ Halg( ¯φ, π(U1)), (7.3)
Halg(φH, U∩ H) ≥ Halg(φH, U2∩ H). (7.4)
From Equations(7.2),(7.3) and(7.4),itfollowsthat
halg(φ)≥ Halg(φ, U )≥ Halg( ¯φ, π(U1)) + Halg(φH, U2∩ H).
Takingthesupremaover U1, U2∈ N (G), weconcludebyapplying Corollary3.5. 2
Wearenow inpositionto provetheAdditionTheorem.
Proof of Theorem7.1. In view of Proposition 7.5, it only remains to prove that halg(φ) ≤ halg( ¯φ) +
halg(φH ).
Let O be anarbitrary opennormalsubgroup of G such thatφ−1(O)≤ O and[O : φ−1(O)] <∞. Then
φ−1(O) isnormalin G (so inparticularin O itself), and
O
φ−1(O)(O∩ H)
φ−1(O) O∩ H
φ−1(O)∩ H
H
Firstobserve that sinceH is φ-stable with ker φ ≤ H,then we alsohave φ−1(H)= H by Lemma7.4, so
φ−1(O)∩ H = φ−1(O)∩ φ−1(H) = φ−1(O∩ H).
Then,computing thesecond indexintherighthandside ofEquation(7.5) weobtain
[φ−1(O)(O∩ H) : φ−1(O)] = [O∩ H : φ−1(O)∩ H] = [O ∩ H : φ−1(O∩ H)].
Note that H is a strongly compactly covered group, and having an open normal subgroup O ∩ H
such that φ−1(O ∩ H) ≤ O ∩ H, and [O∩ H : φ−1(O ∩ H)] < ∞ by Equation (7.5). By Proposi-tion5.4,
halg(φH)≥ log[O ∩ H : φ−1(O∩ H)]. (7.6)
Tocomputethefirstindexintherighthandsideof Equation(7.5),firstnote that
φ−1(O)(O∩ H) = (φ−1(O)H)∩ O
O
φ−1(O)(O∩ H) = (φ−1(O)H)∩ O
φ−1(O) O∩ H
H π(H)
φ−1(O)H π(φ−1(O)) = ¯φ−1(π(O))
OH π(O)
G π G/H
of thesubgroups of G and G/H, onecaneasilyverifythat [O : φ−1(O)(O∩ H)] = [O : (φ−1(O)H)∩ O] = [OH : φ−1(O)H] =
= [π(OH) : π(φ−1(O)H)] = [π(O) : π(φ−1(O))] = [π(O) : ¯φ−1(π(O))].
Byour assumption,G/H is astrongly compactlycovered group.As π(O) is an opennormal subgroup of G/H, suchthat φ¯−1(π(O)) ≤ π(O) and [π(O): ¯φ−1(π(O))] < ∞ byEquation (7.5), Proposition 5.4 appliedto G/H and to φ implies ¯ that
halg( ¯φ)≥ log[π(O) : ¯φ−1(π(O))]. (7.7)
SummingupEquation(7.6) andEquation(7.7),and usingEquation(7.5),weobtain
halg( ¯φ) + halg(φH)≥ log[O : φ−1(O)].
By the arbitrariness of O and applying Proposition 5.4 to G we conclude that halg( ¯φ) + halg(φ H) ≥
halg(φ). 2
Remark7.6.ForthesubclassofcompactlycoveredlocallycompactabeliangroupsTheorem7.1admitsan alternativeproof.Indeed,onecancombinetheAdditionTheoremprovedin[21] forthetopologicalentropy of continuous endomorphisms of locally compact totally disconnected groups with the Bridge Theorem provedin[10] (seeTheorem 6.5).
Thenextcorollaryshowsthatwhenwecomputethealgebraicentropyofatopological automorphism of
astronglycompactlycoveredgroup G, wemayassumethat G is alsototallydisconnected,i.e.,itsconnected component c(G) is trivial.
Corollary 7.7. Let G be a strongly compactly covered group, and φ ∈ End(G) be such that c(G) is φ-stable and contains ker φ. Then halg(φ)= halg( ¯φ), where φ : G/c(G) ¯ → G/c(G) is the induced map.
In particular, if φ ∈ Aut(G), then c(G) is φ-stable and halg(φ)= halg( ¯φ).
Proof. As G is compactly covered by Proposition 1.1, the connected component c(G) is compact by [2, Lemma2.15],so halg(φc(G))= 0.ApplyingTheorem 7.1weget
halg(φ) = halg( ¯φ) + halg(φc(G)) = halg( ¯φ).
Forthelastassertionusethefactthat c(G) is acharacteristic subgroupof G. 2
Acknowledgements
Itisapleasureto thankDikranDikranjanfortheinterestingdiscussionsonthefirstdraftofthispaper. This work is supported by Programma SIR 2014 by MIUR,project GADYGR, number RBSI14V2LI, cupG22I15000160008.
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